A correct definition of sequential right continuity of a function

[psie]

TL;DR Summary

In measure theory, if $\mu$ is a finite Borel measure on the real line, then $F(x)=\mu((-\infty,x])$ defines an increasing and right continuous function from $\mathbb R\to\mathbb R$. I realize I don't have any firm definition at hand to check that it is right continuous (nor have I found any definition online). Hence let me present one.

Here's my definition I've been working on.

Let $f:A\subset\mathbb R\to\mathbb R$ be given. If $c\in\mathbb R$ is a limit point of $A^+=\{x\in A:x\ge c\}$, then $f$ is right continuous at $c$ iff $$\lim_{n\to\infty}f|_{A^+}(x_n)=f|_{A^+}(c),$$ for every sequence $(x_n)$ in $A^+$ such that $x_n\to c$ as $n\to\infty$ and where $f|_{A^+}$ is the restriction of $f$ to $A^+$.

Comments? Suggestions for improvements?

EDIT: The reason I'm looking for a sequential characterization of right continuous is because the way you check that $F$ is right continuous is through $$F(x)=\mu((-\infty,x])=\mu\left(\bigcap_{n=1}^{\infty}(-\infty,x_n]\right)=\lim_{n\to\infty}\mu((-\infty,x_n])=\lim_{n\to\infty}F(x_n),$$for any sequence $(x_n)$ such that $x_n\searrow x$ as $n\to\infty$.

 

[fresh_42]

Hewitt/Stromberg define it that way (Def. 8.18, p. 111) for complex functions on the real line but resolve the question about the meaning of the limit by classical epsilontic in their chapter about Riemann/Stieltjes. They also define continuous and purely discontinuous measures but I couldn't find a definition that used measures instead of norms for functions. Of course, you can use measures to define convergence, but that's not the continuity problem, only the meaning of the limit in the definition.